Problem: There exists a constant $k$ so that the minimum value of
\[4x^2 - 6kxy + (3k^2 + 2) y^2 - 4x - 4y + 6\]over all real numbers $x$ and $y$ is 0.  Find $k.$
Solution: We can write the expression as
\begin{align*}
4x^2 - 6kxy + (3k^2 + 2) y^2 - 4x - 4y + 6 &= x^2 - 4x + 4 + 2y^2 - 4y + 2 + 3x^2 - 6kxy + 3k^2 y^2 \\
&= (x^2 - 4x + 4) + 2(y^2 - 2y + 1) + 3(x^2 - 2kxy + k^2 y^2) \\
&= (x - 2)^2 + 2(y - 1)^2 + 3(x - ky)^2.
\end{align*}The only way that this expression can take on the value of 0 is if $x = 2,$ $y = 1,$ and $x = ky.$  Thus, $k = \boxed{2}.$